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Bell Work: Given the sets L = {0, 1, 2, 3}, M = {5, 6, 7}, and N = {0, 1}, are the following statements true or false? (a) 6 L (b) 0 N Answer: (a) False (b) True Lesson 61: Subsets, Subsets of the Set of Real Numbers If all the members of one set are also members of a second set, the first set is said to be a subset of the second set. If we have the two sets B = {1, 2} A = {1, 2, 3} Then we can say that set B is a subset of set A because all the members of set B are also members of set A. We use the symbol to mean is a subset of. Therefore, we can write B A Which is read as “set B is a subset of set A” Since there are members of set A which are not members of set B, we say “set B can be properly contained in set A” or simply “set B is a proper subset of set A.” The slash can be used to negate the symbol . We can write A B Which is read “set A is not a subset of set B” because all the members of set A are not members of set B The set that has no members is defined to be a proper subset of every set that has members and to be an improper subset of itself. This set is called the empty set or the null set and can be designated by using either of the symbols shown here. { } empty set null set Example: Given the sets D = {0, 1, 2}. E = {1, 2, 3, …}, and G = {1, 3, 5}, tell which of the following assertions are true and which are false and why. (a) E G (b) G E (c) D E Answer: (a) False, all members of set E are not members of set G. (b) True, All members of set G are members of set E (c) False, Zero, which is an element of set D, is not a member of set E. Real Numbers Rational Numbers Real numbers that can be expressed as the fraction of two integers: ½, 0.3, -2/3 etc. Irrational Numbers Real numbers that cannot be expressed as the fraction of two integers, such as: √5, √2. -√2, 2√2, π, 3π, etc. Integers Set of whole numbers and the Opposites: …-1, 0, 1, … Whole Numbers Set of natural numbers & 0: 0, 1, 2, 3, 4, … Natural or Counting Numbers 1, 2, 3, 4, …. Symbols to be aware of. Useful facts about irrational numbers that can be used in problem sets. 1. The product of an irrational number and a nonzero rational number is an irrational number. 2. The sum of a rational number and an irrational number is an irrational number. Example: ½ {What subsets of the real numbers}? Answer: This asks that we identify the subsets of the real numbers of which the number ½ is a member. Rational and Real Numbers Practice: 5 {What subsets of real numbers}? Answer: Natural, Whole, Integer, Rational, and Real Practice: 3√2 {What subsets of real numbers}? Answer: Irrational Numbers and Real Numbers Practice: Tell whether the following statements are true or false and then explain why: (a) {Reals} {Integers} (b) {Irrationals} {Reals} (c) {Irrationals} {Rationals} (d) {Wholes} (e) {Integers} {Naturals} {Reals} Answer: (a) False (b) True (c) False (d) False (e) True HW: Lesson 61 #1-30